Given two arrays arr[] containing N integers and Q[][] containing K queries where every query represents a range [L, R]. The task is to rearrange the array and find the maximum possible sum of all the subarrays where each subarray is defined by the elements of the array in the range [L, R] given by each query.
Note: 1-based indexing is used in the Q[][] array to signify the ranges.
Examples:
Input: arr[] = { 2, 6, 10, 1, 5, 6 }, Q[][2] = {{1, 3}, {4, 6}, {3, 4}}
Output: 46
Explanation:
One possible way is to rearrange the array to arr[] = {2, 6, 10, 6, 5, 1}.
In this arrangement:
The sum of the subarray in the range [1, 3] = 2 + 6 + 10 = 18.
The sum of the subarray in the range [4, 6] = 6 + 5 + 1 = 12.
The sum of the subarray in the range [3, 4] = 10 + 6 = 16.
The total sum of all the subarrays = 46 which is the maximum possible.
Input: arr[] = { 1, 2, 3, 4, 5, 6, 7, 8 }, Q[][2] = {{1, 4}, {5, 5}, {7, 8}, {8, 8}}
Output: 43
Explanation:
One possible way is to rearrange the array to arr[] = {2, 3, 4, 5, 6, 1, 7, 8}.
In this arrangement:
The sum of the subarray in the range [1, 4] = 2 + 3 + 4 + 5 = 14.
The sum of the subarray in the range [5, 5] = 6 = 6.
The sum of the subarray in the range [7, 8] = 7 + 8 = 15.
The sum of the subarray in the range [8, 8] = 8 = 8.
The total sum of all the subarrays = 43 which is the maximum possible.
Approach: On observing clearly, one conclusion which can be made is that we get the maximum sum when the maximum elements are included in as many subarrays as possible. For this, we need to find the number of times every index is included by iterating all the queries.
For example: Let the array be arr[] = {2, 6, 10, 6, 5, 1} and the queries be Q[][] = {{1, 3}, {4, 6}, {3, 4}}.
- Step 1: Create a count array C[] of size N. So, initially, the count array C[] = {0, 0, 0, 0, 0, 0}.
- Step 2: For the query [1, 3], the elements at the index [1, 3] are incremented by 1. The count array after this query becomes {1, 1, 1, 0, 0, 0}.
- Step 3: Similarly, for the next query, the count array becomes {1, 1, 1, 1, 1, 1} and finally, after the third query, the count array becomes {1, 1, 2, 2, 1, 1}.
- Step 4: After obtaining the count array, the idea is to use sorting to get the maximum sum.
- Step 5: After sorting, the array C[] = {1, 1, 1, 1, 2, 2} and arr[] = {1, 2, 5, 6, 6, 10}. The maximum possible sum is the weighted sum of both the arrays, i.e.:
sum = ((1 * 1) + (1 * 2) + (1 * 5) + (1 * 6) + (2 * 6) + (2 * 10)) = 46
Below is the implementation of the above approach:
C++
// C++ program to find the maximum sum// after rearranging the array for K queries#include <bits/stdc++.h>using namespace std;// Function to find maximum sum after// rearranging array elementsint maxSumArrangement(int A[], int R[][2], int N, int M){ // Auxiliary array to find the // count of each selected elements int count[N]; // Initialize with 0 memset(count, 0, sizeof count); // Finding count of every element // to be selected for (int i = 0; i < M; ++i) { int l = R[i][0], r = R[i][1] + 1; // Making it to 0-indexing l--; r--; // Prefix sum array concept is used // to obtain the count array count[l]++; if (r < N) count[r]--; } // Iterating over the count array // to get the final array for (int i = 1; i < N; ++i) { count[i] += count[i - 1]; } // Variable to store the maximum sum int ans = 0; // Sorting both the arrays sort(count, count + N); sort(A, A + N); // Loop to find the maximum sum for (int i = N - 1; i >= 0; --i) { ans += A[i] * count[i]; } return ans;}// Driver codeint main(){ int A[] = { 2, 6, 10, 1, 5, 6 }; int R[][2] = { { 1, 3 }, { 4, 6 }, { 3, 4 } }; int N = sizeof(A) / sizeof(A[0]); int M = sizeof(R) / sizeof(R[0]); cout << maxSumArrangement(A, R, N, M); return 0;} |
Java
// Java program to find the maximum sum// after rearranging the array for K queriesimport java.util.*;class GFG{ // Function to find maximum sum after // rearranging array elements static int maxSumArrangement(int A[], int R[][], int N, int M) { // Auxiliary array to find the // count of each selected elements int count[] = new int[N]; int i; // Finding count of every element // to be selected for ( i = 0; i < M; ++i) { int l = R[i][0], r = R[i][1] + 1; // Making it to 0-indexing l--; r--; // Prefix sum array concept is used // to obtain the count array count[l]++; if (r < N) count[r]--; } // Iterating over the count array // to get the final array for (i = 1; i < N; ++i) { count[i] += count[i - 1]; } // Variable to store the maximum sum int ans = 0; // Sorting both the arrays Arrays.sort( count); Arrays.sort(A); // Loop to find the maximum sum for (i = N - 1; i >= 0; --i) { ans += A[i] * count[i]; } return ans; } // Driver code public static void main(String []args) { int A[] = { 2, 6, 10, 1, 5, 6 }; int R[][] = { { 1, 3 }, { 4, 6 }, { 3, 4 } }; int N = A.length; int M = R.length; System.out.print(maxSumArrangement(A, R, N, M)); }}// This code is contributed by chitranayal |
Python3
# Python3 program to find the maximum sum # after rearranging the array for K queries # Function to find maximum sum after # rearranging array elements def maxSumArrangement( A, R, N, M): # Auxiliary array to find the # count of each selected elements # Initialize with 0 count = [0 for i in range(N)] # Finding count of every element # to be selected for i in range(M): l = R[i][0] r = R[i][1] + 1 # Making it to 0-indexing l = l - 1 r = r - 1 # Prefix sum array concept is used # to obtain the count array count[l] = count[l] + 1 if (r < N): count[r] = count[r] - 1 # Iterating over the count array # to get the final array for i in range(1, N): count[i] = count[i] + count[i - 1] # Variable to store the maximum sum ans = 0 # Sorting both the arrays count.sort() A.sort() # Loop to find the maximum sum for i in range(N - 1, -1, -1): ans = ans + A[i] * count[i] return ans# Driver code A = [ 2, 6, 10, 1, 5, 6 ]R = [ [ 1, 3 ], [ 4, 6 ], [ 3, 4 ] ] N = len(A)M = len(R)print(maxSumArrangement(A, R, N, M))# This code is contributed by Sanjit_Prasad |
C#
// C# program to find the maximum sum// after rearranging the array for K queriesusing System;class GFG{ // Function to find maximum sum after // rearranging array elements static int maxSumArrangement(int []A, int [,]R, int N, int M) { // Auxiliary array to find the // count of each selected elements int []count = new int[N]; int i; // Finding count of every element // to be selected for ( i = 0; i < M; ++i) { int l = R[i, 0], r = R[i, 1] + 1; // Making it to 0-indexing l--; r--; // Prefix sum array concept is used // to obtain the count array count[l]++; if (r < N) count[r]--; } // Iterating over the count array // to get the readonly array for (i = 1; i < N; ++i) { count[i] += count[i - 1]; } // Variable to store the maximum sum int ans = 0; // Sorting both the arrays Array.Sort( count); Array.Sort(A); // Loop to find the maximum sum for (i = N - 1; i >= 0; --i) { ans += A[i] * count[i]; } return ans; } // Driver code public static void Main(String []args) { int []A = { 2, 6, 10, 1, 5, 6 }; int [,]R = { { 1, 3 }, { 4, 6 }, { 3, 4 } }; int N = A.Length; int M = R.GetLength(0); Console.Write(maxSumArrangement(A, R, N, M)); }}// This code is contributed by Princi Singh |
Javascript
<script>//Javascript program to find the maximum sum// after rearranging the array for K queries//function to sort a arrayfunction arrSort(a, n) { var i, j, min, temp; for (i = 0; i < n - 1; i++) { min = i; for (j = i + 1; j < n; j++) if (a[j] < a[min]) min = j; temp = a[i]; a[i] = a[min]; a[min] = temp; }}// Function to find maximum sum after// rearranging array elementsfunction maxSumArrangement(A, R, N, M){ // Auxiliary array to find the // count of each selected elements var count = new Array(N); // Initialize with 0 count.fill(0); // Finding count of every element // to be selected for (var i = 0; i < M; i++) { var l = R[i][0], r = R[i][1] + 1; // Making it to 0-indexing l--; r--; // Prefix sum array concept is used // to obtain the count array count[l]++; if (r < N) count[r]--; } // Iterating over the count array // to get the final array for (var i = 1; i < N; ++i) { count[i] += count[i - 1]; } // Variable to store the maximum sum var ans = 0; // Sorting both the arrays count.sort(); arrSort(A,N); // Loop to find the maximum sum for (var i = N - 1; i >= 0; --i) { ans += A[i] * count[i]; } return ans;}var A = [ 2, 6, 10, 1, 5, 6 ];var R = [ [ 1, 3 ], [ 4, 6 ], [ 3, 4 ] ];var N = A.length;var M = R.length;document.write( maxSumArrangement(A, R, N, M));//This code is contributed by SoumikMondal</script> |
46
Time Complexity: O(N* log(N))
Auxiliary Space: O(N), where N is the size of the given array.
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